By Frank Ayres Jr.

ISBN-10: 9684229178

ISBN-13: 9789684229174

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Isabelle/Isar — a versatile environment for human-readable formal proof documents. PhD thesis, Institut f¨ ur Informatik, TU M¨ unchen (2002) Towards Constructive Homological Algebra in Type Theory Thierry Coquand1 and Arnaud Spiwack2 1 2 G¨ oteborg University Ecole Normale Sup´erieure de Cachan Abstract. This paper reports on ongoing work on the project of representing the Kenzo system [15] in type theory [11]. Introduction This paper reports on ongoing work on the project of representing the system Kenzo [15] in type theory.

Context export destructs the logical diﬀerence of two contexts, by imposing it on local results. The eﬀect of export Γ 1 Γ 2 is to discharge portions of the context on terms and theorems as follows: export (Γ + fix x ) Γ (t x ) = (λx . t x ) export (Γ + fix x ) Γ ( B x ) =( x. B x) export (Γ + assume A) Γ (A B ) = ( A =⇒ B ) Discharging assumptions on syntactic terms has no eﬀect. 2 Context Data Isabelle belongs to the tradition of “LCF-style” provers, which is centered around formally checked entities being implemented as abstract datatypes.

1 (A, C) = Prove(A, C) is deﬁned iﬀ A is a set of syntactic formulas in Loop and C is a formula in Loop . It returns a syntactic representation of a proof if one exists. 2. π ˆ2 is the theorem proving process performed by some automated theorem prover. It takes the elements of A as assumptions and C as a conclusion. 3. M is P roves(Prove(A, C), A C) Here the predicate P roves checks the correctness of the derivation. The formalisation of this predicate depends on the calculus of the integrated prover and is generally very lengthy to formalise.

### Algebra moderna by Frank Ayres Jr.

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